 Now we're often interested in relating rainfall, the amount of rainfall that comes down, to the amount of runoff that's flowing off. And perhaps that runoff might be in a stream. For example, here I have an example of a hydrograph where we're plotting the time t on the horizontal axis and our flow rate, q, on the vertical axis. And one of the things we're interested in, perhaps, is the peak flow rate. The maximum amount of flow that's going by at any period in time. So if we're interested in doing that, we're trying to figure out how much runoff is coming off. There's a very simple method of doing this called the rational method. And what this rational method does is it relates the discharge q to the intensity of the rainfall using a very simple equation. q, there's our peak discharge, is equal to some constant c times the rainfall intensity i times the area a, and this area is the area of the watershed or the area that we're considering that the rain is falling on. It could be something as large as the watershed, although generally it's applied to smaller areas. For example, something like a parking lot or a particular region, maybe in a housing development or something along those lines. So again, that area is if we're thinking about the rainfall falling on some two-dimensional area with a particular intensity. And now the intensity we're talking about is a measurement we've seen a little bit before, which is usually a measurement of inches per hour. It's some measurement of depth of rainfall fall in some units of time, inches per hour. And then again, our area is going to be in some sort of area, maybe acres, maybe feet, it might be some square feet or something along those lines, inches per hour, square feet. And notice if we do that, we're going to get a flow rate in volume, feet cubed times inches, or feet squared times inches would be cubic feet if we do the right conversions, volume over time. So this is going to end up being something in volume over time. And then we simply have this number here called the runoff coefficient. And the idea here is that we're simply thinking about the amount of rain, the volume of rain here represented by IA, and we're going to multiply it by some fraction. We're going to assume only some fraction of it runs off. And that's effectively what this coefficient is going to represent is that fraction, what percentage of it's going to run off. For example, let's say we had a plot of land that has an area of 2.4 acres. We have some sort of 2.4 acre plot of land. And a storm comes through, we'll call it a 25-year storm, something we assume every 25 years comes through. Storm comes through and we'd like to know what our peak runoff is going to come from this 25-year storm. And we look up our intensity duration frequency curves and we find out that a 25-year storm in a particular area might have an intensity of 3.6 inches per hour. Well, what I'm going to do is I'm going to consider the volume, the total volume that's going to fall on this particular area is going to be equal to that intensity times the area. So V equals IA, volume equals intensity times area. So we multiply our 3.6 inches per hour times the area of 2.4 acres. And now we're going to need to do some conversions to actually get an appropriate set of units, 43,560 feet in an acre. Okay? As well as conversions for one foot is 12 inches and one hour being 3,600 seconds if we actually want to do this. And if we do that calculation, we multiply all those numbers out, we might get something like 8.7 cubic feet per second or 8.7 CFS, cubic feet per second. Now notice this value here is the total amount of rainfall, but only some fraction of that rainfall ends up in runoff and ends up in runoff in the period of time necessary to contribute to this peak value. So we think about that as being some fraction. And what we do is we, this is actually an empirical relationship. What we do is we go look up on some charts. There are numerous charts that give values for this runoff coefficient. And they have lots of dependencies. They depend upon things like the slope of the area. They depend upon the amount of vegetation. They depend upon the amount of pervious surface. For example, I have a little chart over here, okay, of a few examples of land use as compared to the coefficients. And in this particular case, I'm sorry, this C should be C sub I as in, we'll just leave it as C for the moment. Okay, let's say for example, we have apartments on this 2.4 acre area and we want to know what the peak overflow is going to be. Well, we would simply say that our Q would be equal to 0.60 times that IA, which is the 8.7 CFS. And that would give us a value of 5.22 cubic feet per second as an estimate of this peak flow here. Notice that's very different than if we have this falling on a very large 2.4 acre field that's effectively a lawn that's maintained on flat lawn, maintained on sandy soil. In that case, the coefficient has been established as being a much smaller value of 0.08. If I plug that value in instead, 0.08 and multiply it, I get a much smaller value of 0.70 cubic feet per second. Now, we can apply the rational method even if we don't have a consistent classification of the area that we're talking about. For example, if the area you're looking at is not entirely apartments or not entirely flat lawn, we can still apply this rational method by dividing up the area we're interested in into different groups or categories. For example, you might recognize that a certain piece might have a particular classification. For example, this might be open space which might abut to some forest. You might have part of the area that might be residential. Maybe there's some commercial use over here. And I'm giving some example. Now, I've done a poor job here in that I probably, maybe this is also forest. And then some fraction in between might represent streets. And the goal here is if you're able to associate areas, parts of the area with each of these categories. For example, the forest might have a total amount, total area. The forest might have a given total area that we, oops, I'm having problems with my markers here. There might be some area associated with the forest. Maybe we can call that A1, some area associated with the open space A2, A3, A4, and the streets A5. If we can find the different areas associated with each of our different categories, well, then what we can do is we can use area waiting to find a, we can use area waiting to find a coefficient that takes into account all the different surfaces we have. But we'll need to do this by counting them based on how much area each takes up. For example, let's consider this space that I sort of sketched here and let's assume that there are some waiting, there's a different number of acre associated with each of these types of land use. For example, I'll make a list over here, AI. And we have these values, CI, where I is going to be the index for each of them, maybe open space is one, forest is two, residential is three, and so on and so forth. So we have the different coefficients, and let's assume that we measure the acreage associated with each of these. For example, 14.2 acres of open space, 11.6 acres of forest, 8.9 acres of residential, 4.3 acres of light commercial, and 3.9 acres of street. So there are different values associated with each of these different areas. What we're going to do to define the C that we're going to use here, this coefficient, is we're going to create a weighted average. We're going to take each coefficient, I, the list of coefficients that corresponds with each type, we're going to multiply it by the area that's associated with it, and then we're going to sum all of those values. After we've done that, we're going to normalize again by taking the sum of all the areas dividing out by the total area. In other words, you can think of each of these pieces as being a fraction that it contributes. The forest contributes A1 over the total A as a fraction of the coefficient. If I do this calculation here, then I will get a weighted average coefficient. That coefficient is the one that we can then plug in here and use the overall area and the overall rainfall intensity to estimate the peak discharge. For example here, I would take each of these, I would take the 14.2 acres times the 0.19 acres. I would add the 11.6 acres, multiplied by the 0.14 value coefficient, plus the 8.9 acres times the 0.32, etc. I recognize that these coefficients don't have any units, but this entire thing is going to be in units of acres. Then when I divide that entire value, it's going to be the total of all the area, which we'll see is 42.9 acres if you sum up all this total amount of area here. When you do that, it's going to cancel out acres and the value you get is unitless, which corresponds with our unitless coefficient. If I do that in this case, I get a value C equals 0.33. Notice just to make sure that this makes sense, we look and we see that that 0.33 is sort of in the middle here. It's definitely not the same as our low value, our lower two values here, but it does tend to be closer to our lower values than our high values here. But you'll notice that makes sense because the low values are more heavily represented in area. There is more area of open space and forest than there is of these values for light commercial and the streets, the things that are going to tend to have more of the runoff. So what this says is roughly a third of the overall volume that falls on this particular area will contribute to the peak discharge here.